Abstract: After a brief introduction to the history of the subject, I will motivate the conjecture that a smooth plane sextic curve cannot have more than 72 tritangents, i.e., lines intersecting the curve with even multiplicity at each point. (A stronger conjecture is that the number of tritangents is 72 or at most 68, with all values taken.) I will also put the problem into a larger context and discuss the known results and a few steps towards the proof of this conjecture.